Structure Connectivity and Substructure Connectivity of $k$ -Ary $n$ -Cube Networks

Abstract

We present new results on the fault tolerability of k-ary n-cube (denoted Q k n ) networks. Q k n is a topological model for interconnection networks that has been extensively studied since proposed, and this paper is concerned with the structure/substructure connectivity of Q k n networks, for paths and cycles, two basic yet important network structures. Let G be a connected graph and T a connected subgraph of G.The T -structure connectivity κ(G; T ) of G is the cardinality of a minimum set of subgraphs in G, such that each subgraph is isomorphic to T , and the set's removal disconnects G. The T -substructure connectivity κ s (G; T ) of G is the cardinality of a minimum set of subgraphs in G, such that each subgraph is isomorphic to a connected subgraph of T , and the set's removal disconnects G. In this paper, we study κ(Q k n ; T ) and κ s (Q k n ; T ) for T = P i , a path on i nodes (resp. T = C i , a cycle on i nodes). Lv et al. determined κ(Q k n ; T ) and κ s (Q k n ; T ) for T ∈ {P 1 , P 2 , P 3 }. Our results generalize the preceding results by determining κ(Q k n ; P i ) and κ s (Q k n ; P i ). In addition, we have also established κ(Q k n ; C i ) and κ s (Q k n ; C i ).

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